Edgar Allan Poe’s Riddle:Framing Effects in Repeated Matching Pennies Games*

Kfir EliazDepartment of Economics, Brown University

Ariel RubinsteinThe University of Tel Aviv Cafés and New York University

April 2009

AbstractFraming effects have a significant influence on the finitely repeated matching penniesgame. The combination of being labelled "a guesser", and having the objective ofmatching the opponent’s action, appears to be advantageous. We find that being aplayer who aims to match the opponent’s action is advantageous irrespective ofwhether the player moves first or second. We examine alternative explanations for ourresults and relate them to Edgar Allan Poe’s "The Purloined Letter". We propose abehavioral model which generates the observed asymmetry in the players’performance.

JEL Classification: C91, C72Keywords: Matching pennies, Even and odd, Framing effects, Edgar Allan Poe

*We thank Marina Agranov for help in running the experiments, Eli Zvuluny forprogramming, Yoav Benyamini for advising us on statistical matters, Yaniv Ben-Amiand Zhauguo Zhhen for their assistance in the data analysis and John Wooders andJörg Stoye for helpful comments. We wish to especially thank the Associate editor ofthis journal and three referees for their exceptional comments which helped improvethe paper. Financial and technical support from the Center for Experimental SocialScience at New York University is gratefully acknowledged.

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1. Introduction

In his short story “The Purloined Letter”, Edgar Allan Poe writes,

"I knew one about eight years of age, whose success at guessing in thegame of ‘even and odd’ attracted universal admiration. This game issimple, and is played with marbles. One player holds in his hand a numberof these toys, and demands of another whether that number is even or odd.If the guess is right, the guesser wins one; if wrong, he loses one. The boyto whom I allude won all the marbles of the school. Of course he had someprinciple of guessing; and this lay in mere observation and admeasurementof the astuteness of his opponents. For example, an arrant simpleton is hisopponent, and, holding up his closed hand, asks, ‘are they even or odd?’Our schoolboy replies, ‘odd,’ and loses; but upon the second trial he wins,for he then says to himself, ‘the simpleton had them even upon the firsttrial, and his amount of cunning is just sufficient to make him have themodd upon the second; I will therefore guess odd;’ - he guesses odd, andwins. Now, with a simpleton a degree above the first, he would havereasoned thus: ‘This fellow finds that in the first instance I guessed odd,and, in the second, he will propose to himself, upon the first impulse, asimple variation from even to odd, as did the first simpleton; but then asecond thought will suggest that this is too simple a variation, and finallyhe will decide upon putting it even as before. I will therefore guess even;’ -he guesses even, and wins. Now this mode of reasoning in the schoolboy,whom his fellows termed ‘lucky,’ - what, in its last analysis, is it?’ ‘It ismerely,’ I said, ‘an identification of the reasoner’s intellect with that of hisopponent.’ "

Poe could have assigned the gifted boy either of the two roles: the first mover, whochooses the number of marbles, or the second mover, who guesses whether thatnumber is odd or even. Poe described the boy as the guesser. Our experimentalevidence suggest that this may not have been just a coincidence.

The game Poe describes is essentially a standard (finitely) repeated game in whichthe stage game is the two-player zero-sum game known as "matching pennies":

Player 1

Player 2L R

T 0,1 1,0B 1,0 0,1

Figure 1

All game theoretic solutions we are aware of, predict that each player’s chances of"winning" at each round is 50%. Hence, the win rate of each player is distributed

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according to Binomialn, 0. 5 where n is the number of rounds.Poe, however, subjects the above game to three framing effects. First, he

introduces timing: players move sequentially but without observing each other’sactions. Second, he labels the players: the second mover is referred to as “a guesser”.Third, he labels the actions: one player chooses an odd or even number and anotherannounces “odd” or “even”. A feature of this labeling is that one player wins if hisaction matches that of his opponent, while the other player wins if the two choosedifferent actions. Do these framing effects have any real impact on the behavior ofplayers? In particular, can such framing effects give an advantage to one of theplayers, as Poe seems to suggest in his story?

To address these questions, we conducted a series of laboratory experiments inwhich subjects played different variants of a finitely repeated matching-pennies game.In the baseline treatment (G1), subjects played a simplified version of Poe’s originalgame, in which the stage game displayed the three framing effects mentioned above:(1) players moved sequentially, but the second player did not observe the actionchosen by the first player, (2) the first player is labeled “a misleader” and the secondplayer is labeled “a guesser”, and (3) the actions available to each player are labeled“0” and “1”, such that the guesser wins whenever he chooses the same action as themisleader. Subjects were randomly paired to play 24 rounds of this game (sufficientlymany rounds to allow for learning and not excessive to prevent drift of attention).

Our experimental findings in G1 suggest that the combination of being labelled aguesser, having the objective of matching the opponent’s action and moving second,is advantageous. On average, guessers won about 53% of the rounds, and won morepoints than the misleader in 28 of the 55 participating pairs, while misleaders wonmore points in only 16 of the pairs (11 pairs ended with a "draw"). Hence, theconjunction of all three framing effects tilts the game in favor of the guesser. Theguesser’s advantage is not weakened in the second half of the game, indicating thatlearning does not eliminate it.

We conjecture that each of the above three framing manipulations could be asource of the guesser advantage in G1.

(1) Past studies in psychology have shown that priming of a particular role orpersonality trait can affect subsequent behavior in the direction of the trait or role (seeBargh, Chen and Burrows (1996)). In our context, player 1 is primed to be a misleaderand focus more on creating seemingly random patterns and less on predicting hisopponent’s response. Player 2, on the other hand, is primed to be a guesser andfocus on forecasting his opponent’s action by detecting regularities in the misleaders’past actions. The guesser’s advantage may be traced to the empirical observation thatindividuals are not good at randomizing and are better skilled at detecting patterns(see for example, Bar-Hillel and Wagenaar (1991)). This suggests that whileattempting to randomize, the misleaders create systematic patterns that are partiallydetected by the guesser

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(2) The labels on the actions create an asymmetry between the players: one playerwins if he matches his opponent’s action, while the other player wins if he mismatcheshis opponent action. The notion of stimulus-response compatibility suggests that it iseasier for a subject to match an exogenous stimulus than to mismatch it (see Simon,Sly and Vilapakkam (1981)). In our case, the stimulus is a player’s own guess of hisopponent’s action. We conjecture that in this case as well, the guesser’s task ofmatching the stimulus is easier than the misleader’s task of mismatching it.

(3) The role of timing-without-observability is somewhat of a puzzle to us. Previousstudies have demonstrated its role in selecting equilibria (see the related literaturesection). However, it does not seem that those studies are relevant to a zero-sumgame with a unique Nash equilibrium.

To better understand the contribution of each framing effect, we conducted fourtreatments where subjects played variants of G1 that differed only in the framing of thebasic stage game. In G2 we dropped the players’ labels, misleader and guesser, andframed the one shot game as an "even and odd" game where player 1 is the "odd"player and player 2 is the "even" player (namely, player 2 wins if the sum of thechosen numbers is even). Our data reveals an advantage to the even player: he wonon average 54% of the rounds. This suggests that two of the three framing effects -moving second and having to match the opponent’s action- are enough to generate anadvantage.

In order to check whether the advantage of player 2 in G1 and G2 is actually a"second-mover advantage", we ran two "mirror" treatments: G1*, which is identical toG1 except that it is player 1 who has to guess player 2’s future choice; and G2*, whichis identical to G2 except that player 1 is the "even" player. We don’t find a significantadvantage to the misleader/odd-player when he moves second.

To further explore the role of timing, we ran a treatment, G3, in which we kept thesequential structure of the original game but dropped the non-neutral labels for boththe players and the actions. In G3, the first player chose a letter "a" or "i", while thesecond player chose a letter "s" or "t". Player 1 (player 2) wins whenever the wordwhich is created is either "at" or "is" ("as" or "it"). In this version, where neither primingnor stimulus-response compatibility played a role, we cannot reject the hypothesis thateach player is equally likely to win. This suggests that timing-without-observabilityalone is not enough to generate an advantage to the second mover.

In light of these findings, we propose a behavioral model, which generates theobserved advantage to the player in the role of a guesser/even-player. In this modelthe players use the following stochastic behavioral rule. With some probability theytoss a biased coin to determine whether or not they keep their previous action (thisbias can be in favor or against switching). With the complementary probability, they“think strategically” and respond optimally to the prediction of their opponent, which isbased on his last action. We allow for two asymmetries in this model. First, we allowfor differences in the frequencies by which the two players think strategically. Second,

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we allow for the possibility of a mistake in executing the best response to theprediction. We assume that: (i) the guesser/even player is more likely to thinkstrategically than the misleader/odd-player, and (ii) the misleader/odd-player is morelikely to err in executing his optimal response since it is a more complicated operation(he has to mismatch where the guesser/even player has to match). We find that bothasymmetries, each by itself and to a larger extent together, generate about 52%advantage to the guesser/even-player for a reasonable range of parameters.

Related literatureClosest in spirit to our paper is Rubinstein, Tversky and Heller (1996), which examinesseveral variations of a hide and seek game. In the baseline version, "the hider" has to"put a treasure" in one of four boxes labeled from left to right A-B-A-A, and "theseeker" has the opportunity to search for it in one of the boxes. According to thegame-theoretical prediction of this zero-sum game, both players uniformly randomizebetween the four boxes and thus, the probability that the seeker finds the treasure is25%. In the original research, as well as in subsequent replications of the experiment,central A is the most popular choice among both the hiders and the seekers. Forexample, data collected through the didactic website, gametheory.tau.ac.il, revealsthat the distribution of 1,129 misleaders was (17%, 24%, 36%, 22%) and thedistribution of 1,116 guessers was (10%, 26%, 47%, 16%). According to thesedistributions, the probability of finding the treasure is 28.6%, significantly above theexpected 25%. It seems that subjects in both roles, who wished to play randomly,looked for a choice which is not prominent. They excluded the "distinctive" B and thetwo boxes at the edges, leaving them with the unique "nondistinctive" box, central A.

However, it is not clear whether this result may be interpreted as evidence for theexistence of a guesser’s advantage since this data cannot distinguish between thefollowing alternative explanations: i the normal choice of A is close to 36% and manyguessers think strategically, or ii the normal choice rate of A is closer to 47% andmany of the misleders think strategically. In contrast, we do claim to demonstrate aguesser’s advantage since the data do not point at a prominent action in our setting.

Also related is Attali and Bar-Hillel (2003), which provides field evidence on thedifficulty in preventing guessers from exploiting regularities in supposedly randomsequences of play. The authors looked at high-stakes tests such as SATs anddiscovered that the creators of these tests had a policy of balancing, rather thanrandomizing, the answer keys. This policy was vulnerable to significant exploitation byexaminees. Note, however, that the exam situation is not truly a zero-sum game: thecomposers of the exam want to prevent examinees from guessing too successfully,but they do not want to mislead and fail examinees too often in cases where they donot know the answer.

Repeated matching pennies games were examined in the literature by Mookherjeeand Sopher (1994) and Ochs (1995). In those experiments the matching pennies

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game was presented to the players as a bi-matrix. No significant deviation from themixed strategy equilibrium was reported. Shachat and Wooders (2001) shows that ingames like repeated matching pennies, the Nash equilibrium is the repeated play ofthe stage game equilibrium, even if the players are not risk neutral.

O’Neil (1987) provides a test of the maxmin prediction in a zero-sum game. In hisgame each player has to choose one of the four actions, a,b,c or d, and player 1 winsif the two players choose distinct actions other than d, or if both choose d. Bothplayers’ maxmin mixed strategies are 0.2,0.2,0.2,0.4, and the equilibrium winningprobability of player 1 is 40%. O’Neill claimed that in the finitely repeated game theequilibrium prediction is confirmed. The claim was challenged by Brown andRosenthal (1990) who observed that the data at the individual level reveals significantserial correlations in each player’s choices. According to data collected by one of usvia the site http://gametheory.tau.ac.il, the one shot game behavior is significantlydifferent than the maxmin strategies, 0.16,0.18,0.2,0.45 and 0.14,0.18,0.25,0.44for player 1 and player 2, respectively, though the probability of winning by player 1 is0.403, amazingly close to the equilibrium prediction. In contrast, framing has not effectin the one shot matching pennies.

Several previous experimental studies have demonstrated that timing matters evenwhen two players move sequentially and the early move is unobservable by thesecond player. All of these papers have focused exclusively on the role of timing inselecting among multiple Nash equilibria. Cooper, DeJong, Forsythe and Ross (1993)have shown that in coordination games players tend to coordinate on the equilibriumpreferred by the first mover. Weber, Camerer and Knez (2004) test the notion of“virtual observability”, according to which players behave as if moves were observable,and hence, tend to coordinate on a Nash equilibrium, which is also thesubgame-perfect equilibrium of the sequential game with observable actions. Ourgame has a unique Nash equilibrium and thus, the effect of timing in our experiment iscompletely different than its effect in those previous studies.

Finally, let us also mention the game theoretical models which analyze therepeated matching pennies game using the bounded rationality approach. (See forexample Ben-Porath (1993), Neyman (1997) and Guth, Kareev and Kliemt (2005)).

2. Experimental design

Subjects were recruited from a pool of students at New York University. Theexperiment was conducted at the laboratory of the Center for Experimental SocialScience (C.E.S.S.) at NYU. An even number (at least 8, at most 20) of subjectsparticipated in each session. The computer randomly matched the subjects intoanonymous pairs (i.e., we used a “partners” design). Each pair played a 24-periodrepeated matching pennies game; in each round, player 1 moved first and player 2moved second without observing player 1’s choice. During each round, each playercould see on his screen the choices made by the players in previous rounds and the

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number of points accumulated so far. We examined five versions of the game thatdiffer only in the framing of the game. The Appendix contains the detailed instructionsfor the basic treatment, G1. The instructions for the other treatments are provided inSection 3. In each session all subjects played the same version of the game. At theend of each session the experimenter calculated the rewards: a sum of $5 was paid toa subject for participation and additional 50 cents were given per each win in the stagegame. Thus, the average amount paid to subjects is $11. The entire process tookabout 30 minutes.

The following table outlines the key features of each of the treatments. A fulldescription of each treatment appears in Section 3. In each of the first four games, abold face letter indicates the player who wins if the two actions coincide.

Treatment Player 1 Player 2 1’s actions 2’s actions Player 1 wins if:

G1 Misleader Guesser 0/1 0/1 Player 2 does not guess him correctly

G1* Guesser Misleader 0/1 0/1 He predicts correctly player 2’s choice

G2 odd even 0/1 0/1 The sum of chosen numbers is odd

G2* even odd 0/1 0/1 The sum of chosen numbers is even

G3 - - a/i s/t The word created is "at" or "is"

Table 1: Key features of the five treatments

3. The basic findings

Table 2 summarizes the results of the five treatments. The results of each treatmentare presented in a separate column and include the distribution of points gained by thesecond movers, their win rate, and both the mean and standard deviation of theirscore. The results are compared with the distribution of B24,0.5. The first statisticaltest is the one-sample Kolmogorov-Smirnov, which is used to compare a sample witha reference probability distribution (in our case B24,0.5). The last line presents thep-values generated by the normal test. The null hypothesis in G1, G2 and G3 is thatthe mean score of player 2 is at most 12 whereas in G1* and G2* the hypothesis isthat the mean score is at least 12.

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n55 n29 n29 n26 n27Player 2’s role guesser misleader even odd as/itPlayer 2’s wins B(0.5, 24) G1 G1* G2 G2* G3

0-7 4% 0% 3% 0% 0% 0%8 4% 0% 0% 3% 8% 11%9 8% 7% 7% 3% 8% 7%

10 12% 7% 7% 3% 19% 4%11 15% 15% 14% 21% 27% 11%12 16% 20% 21% 14% 8% 22%13 15% 16% 14% 17% 12% 11%14 12% 11% 21% 10% 8% 22%15 8% 20% 10% 10% 8% 4%16 4% 4% 3% 7% 0% 0%

17-24 4% 0% 0% 10% 4% 7%

p-value (1-sample K-S*) 1.51% 4.14% 7.78% 46.69% 7.26%

Player 2’s win rate 50.0% 52.6% 51.4% 54.2% 47.9% 51.1%Player 2’s mean score 12.00 12.61 12.34 13.00 11.50 12.26(standard deviation) (0.26) (0.4) (0.45) (0.44) (0.51)p-value (normal test) 3.1% 22.4% 1.4% 14.9% 29.1%

*Bootstrapped to correct for discreteness of the distributions.

Table 2: Summary of the results

3.1. The misleader-guesser gameThe baseline treatment is G1:

"Imagine that you are about to play a 24-round game. In the game there aretwo players, a misleader and a guesser. At each round, the misleadermoves first and has to choose a number, 0 or 1. The guesser moves secondand has to guess the misleader’s choice. The guesser gets a point if hisguess is correct and the misleader gets a point if the guesser fails. Eachplayer’s aim is to get as many points as possible. At each round eachplayer can see on the screen the entire history of the game up to thatpoint."

In G1 the guesser has a significant advantage. On average, guessers gained 12.6points, namely, they won 53% of the rounds. In the repeated game, the guesser won

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more points than the misleader in 28 of the 55 games, while the misleader won morepoints in only 16 games (11 repeated games ended with a "draw"). The results aresignificant at the 3.1% level.

To get a better sense of the guesser’s advantage it is helpful to compare thedistribution of our subjects’ points in G1 with the distribution predicted by B24,0.5.This is depicted in Figure 2 below.

G1

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Number of guesser wins

Num

ber o

f gam

es

Expected Observed

Figure 2: Actual vs. predicted distribution of scores in G1

One may think that a success rate of 0.526 is an insignificant deviation from the 0.5prediction. We disagree. First note that even if both players would use independentmixed strategies, which assign probability 0.6 to one of the two actions, the expectedguesser’s win rate would be only 0.52 (0.60.60.40.4). This indicates that subjects’behavior is far from the game theoretical prediction, and therefore, sophisticatedplayers may obtain an even higher win rate. Second, this "slight" advantage of theguesser could become inflated for a variation of the game (resembling the "even andodd" game in Edgar Allan Poe’s "The Purloined Letter", which is discussed later)where the winner is the player who gets first to an advantage of some fixed number ofpoints.

As mentioned previously, G1 introduces three major framing effects to a standardrepeated matching-pennies game: being labelled a guesser/misleader, aiming tomatch/mismatch the opponent’s action and moving second/first. To investigate theextent to which these framing effects may have contributed towards the guesser’sadvantage, we tested several variations of G1. We discuss these in the nextsubsections.

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3.2. The even and odd gameTo control for non-neutral labeling of players’ roles, we ran the following treatment(G2):

You are about to play a 24-round game. There are two players in the game:player 1 and player 2. At each round, player 1 moves first and player 2moves second. Each player has to choose a number, 0 or 1. Player 1 gets apoint if the sum of the chosen numbers is odd. Player 2 gets a point if thesum is even. Each player’s aim is to get as many points as possible. Ateach round each player can see on the screen the entire history of the gameup to that point.

G2 is a modification of G1 in which the players are not labeled as a misleader anda guesser. While player 2 has to simply choose his guess of player 1’s action, player 1has to choose the opposite of what he anticipates player 2 will choose.

Our experimental results hint at an advantage of the even player (player 2) whowon 54% of the games (on average 13 points), a significant deviation from the gametheoretical prediction with p 1.3%. This suggests that two of the framing effects -aiming to match the opponent’s action and moving second - are enough to generate asignificant guesser advantage.

3.3. The order effectThe results of G1 and G2 leave the possibility that what we observe is actually a"second-mover advantage". In order to examine this possibility, we investigated twoother treatments, dual to G1 and G2 in the sense that the roles of the two players areexchanged. G1* is a guesser-misleader game in which player 1 guesses player 2’schoice. G2* is an even and odd game in which player 1 is the even player.

In both dual game, the second mover (the misleader in G1* and the odd-player inG2*) obtains no advantage. In the even and odd game (G2*), the even player who isthere player 1, even obtained an advantage and gained on average 12.5 points,however this advantage is significant with p15% only.

3.4. The "as/it" gameThe results of G1* and G2* still do not address the question of whether timing by itself(i.e., keeping the labeling of actions and roles as neutral as possible) can generate aguesser’s advantage. We address this question in the final treatment (G3) where theorder of moves is the only asymmetry between the two players.

G3

You are about to play a 24-round game. In the game there are two players:player 1 and player 2. At each round, the two players compose a two-letter

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word. Player 1 moves first and has to choose the first letter of the word, aor i. Player 2 moves second and has to choose the second letter of theword, s or t. Player 1 gets a point if the word formed is either at or is.Player 2 gets a point if the word formed is either as or it. Each player’s aimis to get as many points as possible. At each round each player can see onthe screen the entire history of the game up to that point.

The distribution of points among the 27 pairs who played the game was close tothat of the Binomial(24,0.5). In particular, player 2 won only 51.1% of the rounds (331out of 648 rounds), an insignificant advantage. We interpret these findings asevidence that timing-without-observability on its own cannot generate an advantage toone of the players.

To summarize, our results suggest that an appropriate choice of framing can havean asymmetric effect in what may be perceived as a totally symmetric game. Our dataindicates that being a player who aims to match the opponent’s action isadvantageous irrespective of whether the player moves first or second.

4. Searching for explanations of the guesser’s advantage

4.1. Is there a prominent action?The following could be a possible explanation for the guesser’s advantage in G1.Assume there was a prominent action, which is more likely to be chosen by subjects inboth roles. Then given the payoffs of the game, the tendency of subjects to choose theprominent action would give an advantage to the guesser (this explanation is similar tothe one given in Rubinstein, Tversky and Heller (1996) for the seeker’s advantage).

However, according to the data no action is prominent for both players. In the firstround of G1, 38% of the misleaders and 67% of the guessers chose "1". Taking thesenumbers as estimates for the players’ strategies yields a prediction that the guesserwins at each round with probability 0.46, which does not explain the guesser’sadvantage (similar results were obtained for the other treatments).

Furthermore, a prominent action does not show up in the data of the overall play ofthe game (chi-square test yields a p value of 0.8). The following table presents theoverall frequencies of the outcomes in all rounds in G1. The misleader choosesslightly more zeros and the guesser chooses slightly more ones. Using thesefrequencies to estimate mixed strategies for the whole game yields a prediction thatthe guesser wins with probability 0.4996 ( 0.492 0.523 0.508 0.477), significantlybelow the 0.525 frequency of the guesser’s wins in G1.

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Misleader:

Guesser:

0 1

0 27.0 25.2 52.31 22.2 25.5 47.7

49.2 50.8

Table 3: Distribution of one shot game outcomes

Thus, the existence of some prominent action does not explain the guesser’sadvantage.

4.2. Is the guesser’s advantage an artifact of inexperienced players?One might conjecture that the deviations from the game theoretic prediction are limitedto the beginning of the game and that with time, players learn to play the mixedstrategy equilibrium. By this approach, one would expect that the mean scores in thesecond half of the game would be more symmetric than the mean scores in the firsthalf. The data summarized in Table 4 does not confirm this line of thinking. The lastcolumn of the table presents the results of a two-sample K-S tests of the hypothesisthat the distributions of the results in the two halves is drawn from the samedistribution. The test clearly supports this hypothesis.

Game Guesser’s mean score Two-sample K-S* (p value)in the full game in the first half in the second half

G1 12.6 6.2 6.4 0.966G1* 11.7 5.9 5.8 0.875G2 13.0 6.5 6.5 0.859G2* 12.5 6.3 6.2 0.987G3 12.3 6.2 6 0.985

*Bootstrapped to correct for discreteness of the distributions.

Table 4: Mean number of rounds won by the Guesser

4.3. Persistence effectApplying the Wald-Wolfowitz runs test to the individual data does not reject the

hypothesis that a vast majority of the subjects do produce a random sequence. In G1,for example, the hypothesis of a random sequence is not rejected (p5%) regarding87% of players 1 and 76% of players 2.

One might think that we would observe a "negative recency effect", as reported inthe psychological literature regarding the production of random sequences (see forexample, Kahneman and Tversky (1972)). According to this literature, whenproducing a random sequence of zeros and ones, people tend to change their choices

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from period to period more than 50% of the time. Our data shows the opposite. Weobserve a tendency for persistence. Players 1 and 2 in G1 repeat their past choices in55% and 57% of the cases, respectively. Similar frequencies (between 53% and60%) are observed in the other versions of the game.

Note that the persistence effect does not explain the guesser’s advantage since itcould be that the players are equally likely to repeat their choices after a misleader’swin as after a guesser’s win.

4.4. A partial explanation: asymmetry in response to past successThe following is a transition matrix of the one shot outcomes in G1:

n1,265 1,1 0,0 1,0 0,1

1,1 35.1% 17.7% 21.1% 26.1%0,0 16.9% 36.1% 24.1% 23.0%1,0 25.0% 28.2% 24.6% 22.1%0,1 25.4% 26.7% 21.0% 27.0%

total 25.5% 27.3% 22.7% 24.6%

Table 4:Transition probabilities of one shot outcomes

The one striking observation is the exceptionally significant tendency forpersistence of the outcomes (1,1) and (0,0) and the low transition probabilities from(1,1) to (0,0) and from (0,0) to (1,1). To shed more light on the transition matrix, let uslook at the following table, which displays the chances that each of the players repeatshis action following a success and following a failure:

Repeats action

Misleader success 51.8%failure 57.7%

Guesser failure 52.6%success 60.7%

Table 5: Persistence rate after success/failure

This table reveals that the guesser tends to repeat his action after success and themisleader tends to retain his past action after failure. Suppose the guesser followed asimple strategy where he repeats his past action with probability 0.61 after successand probability 0.53 after failure, and the misleader repeats his action with probability0.58 after failure and with probability 0.52 after success. Then a guesser’s success willbe followed by another success with probability 0.58 0.61 0.42 0.39 0.52, and amisleader’s success will be followed by an equally likely win by each of the players.Calculating the stationary distribution of the Markovian matrix yields a 51% success

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rate for the guesser, and thus, provides a partial explanation for the guesser’ssuccess.

The above is the best partial explanation we found in the data for the guesser’sadvantage. It should be noted that this explanation does not extend to G2. There,player 1 (the odd player) repeats his action after success with probability 0.63 andafter failure with probability 0.50, whereas player 2 (the even player) repeats his actionafter failure with probability 0.56 and after success with probability 0.53. Thesenumbers would be more consistent with a slight advantage for player 1, while in G2we observe an advantage for player 2.

4.5. One more curious factIn G1, there were 73 histories in which the same outcome, (1,1) or (0,0), was repeatedin the last k ≥ 3 periods. In 55% of those cases the guesser succeeded also in thenext period. Strikingly, there were only 28 histories in which the same outcome, (1,0)or (0,1), was repeated in the last k ≥ 3 periods. Such histories were followed by yetanother victory for the misleader in only 36% of the cases. This asymmetry betweenthe players may reinforce our intuition that guessers were more alert than misleaders.

5. Modeling the players’ behaviorThere are two potential asymmetries that may explain the guesser’s/even-player’s

advantage.(i) The misleader/odd-player has to execute a more complicated mental operation.

The guesser/even-player has to predict his opponent’s move and to choose the sameaction. In contrast, the misleader/odd-player has to predict his opponent’s move and tochoose the opposite, an operation which is more likely to yield a mistake.

(ii) The guesser/even-player is more primed than the misleader/odd-player to thinkstrategically, namely to infer the next move from the last move and to respondoptimally to his prediction.

The following model attempts to capture the above asymmetries. Consider arepeated matching-pennies game, where it is player 2 who wins a stage game if thechoices match. Each player chooses his initial action at random. From then on, playeri’s strategy is given as follows. With probability 1 − i he randomizes and withprobability i he "thinks strategically". When a player randomizes he has a bias. Hislast action can predict his next action with probability 1

2 . Here we assume that aplayer repeats his previous action with probability and switches an action withprobability 1 − . If we were to assume that a player is more likely to switch an actionwhen he tries to randomize (what seems consistent with other experiments), ourresults will not change, provided that we also adjust the second (strategic) part ofplayer’s behavioral rule. When a player “thinks strategically” he believes that it ismore likely that his opponent will repeat his last action and he best responds to thisprediction. However, while player 2 executes the best response accurately, player 1makes a mistake with probability , and instead of mismatching his opponent’s action

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he matches it.Assume first that the matching-pennies game is played for infinitely many periods.

There are two states: either the players match thei actions, or they mismatch.Computing the Markov stationary probabilities for these two states, yields the followingexpression for player 2’s win rate:

1−111−21−21−1−211−11−11−111−21−21−1−211−11−−1−1121−2−1−21−1−11−11−

To better understand the relation between this expression and our data, assumenext that the matching-pennies game was repeated for only 24 periods. Assume alsothat 0.6. We simulated the above model to test whether it can generate theguesser/even-player’s advantage observed in our data. Figure 3 plots player 2’s winrate as a function of for 1 2 0.3 (the grid on this graph is of 1% and each pointaverages 10,000 runs of the repeated game).

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

Player One error probability

The effects of Player One error probability

Player Two round win rate

Figure 3: Player 2’s win rate as a function of for 1 2 0.3

As evident from the figure, the asymmetry in the error’s rate (no error for player 2and positive error rate for player 1) is capable of explaining the guesser/even-player’sadvantage: when player 1’s error lies in the interval 0.2,0.3, player 2 gets a lead of1% to 2% above player 1’s win rate. Figure 4 demonstrates that the effect of thisasymmetry is amplified when we consider a player a winner in the repeated game if hescores more than 12 points. Using the same parameters we get that the differencebetween player 2’s win rate and player 1’s win rate increases quickly with andreaches 10% when the mistake probability is 0.3.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

Player One error probability

The effects of Player One error probability

Player Two game win ratePlayer Two game tie ratePlayer Two game lose rate

Figure 4: Player 2’s win rate in the 24 round repeated game as a function of for 1 2 0.3

Looking for the effect of the priming asymmetry, we simulated the model withoutmistakes ( 0) and with asymmetry in the likelihood of behaving strategically. Figure5 plots player 2’s win rate as a function of 1 when 2 0.3. As evident from thegraph, player 2 has an advantage as long as 1 is below 2. In particular when1 0.1 player 2’s win rate is 52% (which is the rate we obtain experimentally in G1).

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.44

0.45

0.46

0.47

0.48

0.49

0.5

0.51

0.52

0.53

0.54

Beta One

The effects of Beta One

Player Two round win rate

Figure 5: Player 2’s win rate as a function of 1 with 2 0.3 and 0

Combining the two asymmetries yields a greater advantage for player 2. Figure 6plots the win rate of player 2 as a function of 2 when 0.2 and 1 0.1. As we

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can see, player 2 obtains a win rate of 0.52 when 2 is around 0.27.

 

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.49

0.5

0.51

0.52

0.53

0.54

0.55

0.56

0.57

0.58

0.59

Beta Two

The effects of Beta Two

Player Two round win rate

Figure 6: Player 2’s win rate as a function of 2 for 0.2 and 1 0.1

Note that some other variants of our model provide paradoxical results. Consider,for example, another behavioral model for the 24-period repeated matching-penniesgame. As before, player i’s behavioral rule has two components, random andstrategic. When a player randomizes he "uses" a random device, which has asystematic bias either towards the action "1" or towards the action "0"; the deviceyields the outcome 1 with either probability 0.6 or 0.4. The bias is determined at thebeginning of the game and is fixed throughout the 24 rounds. In contrast to the firstmodel, the strategic component is triggered by the behavior of his opponent. When aplayer notices that the other player played the same action twice in a row, he bestresponds to that action. The only asymmetry we incorporate to this model is thatplayer 1, and only player 1, makes a mistake with probability when attempting tobest respond to an action that player 2 repeated twice.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Player One error probability

The effects of Player One error probability

Player Two round win rate

Figure 7: Player 2’s win rate as a function of in the alternative model

Note that for this alternative model it is much more complicated to derive themarkov stationary probabilities (assuming infinitely many repetitions of the stagegame) as there are many possible states. We, therefore simulate Player 2’s win rateas function of player 1’s error rate, assuming for 24 repetitions of the stage game. Theresults are plotted in Figure 7. Surprisingly, player 1’s mistake works in his favor (aslong as 0.6)! This fact has to do with the structure of the game, which createsartificial synchronization between the timing in which the players best respond to eachother. This synchronization does not arise in the first behavioral model where it is rarethat the two players will think strategically in the same period.

6. Concluding remarks

6.1. Back to Edgar Allan PoePoe gives the cunning boy the role of the guesser. The boy distinguishes between anopponent who is simple minded and one who is not. If the opponent is simple minded(in our modern jargon, he uses level zero reasoning), the boy predicts that he will notchange his action after success and thus, by changing his own action, he wins thenext round. If the opponent is not simple minded (i.e., he uses level one reasoning),the boy predicts that he will change his action after success, expecting the boy to dothe same. The cunning boy is even more sophisticated (i.e., he uses level tworeasoning) and therefore beats the level one opponent.

Our findings are related to Poe’s intuition and would fit nicely in our explanation ofsubjects’ behavior after a guesser’s success (rather than a guesser’s failure),assuming that guessers believe that the misleaders are not simple minded. As wedescribed in subsection 4.4, following a misleader’s failure, 57.5% of the misleaders

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and 60.7% of the guessers repeat their action. However, Poe demonstrates the boy’scunningness after his failure where we do not find traces of such thinking.

Note that the game Poe describes is somewhat different from ours. The closestparallel will be a version in which the first player to have an advantage of k points winsthe match. If the guesser’s victories are i.i.d. across games with a winning probabilityof q, then his chances of being the first to have an advantage of k points is

11 1−q

q k

It follows from this formula that guessers’ "minor" advantage in the one shot game istranslated to a huge advantage in Poe’s game. For example, if q 0.526 and k 4 theguesser will win the game with probability 0.6, or if k 8 his chances of winning willjump to 0.7

6.2. Game TheoryMuch has been written about the failures of Game Theory to predict behavior.Advocates of Game Theory question the significance of the experimental evidencewhich reject the game theoretical predictions. Some claim that the refutation wasconducted over games, which the subjects were not experienced to play. Some arguethat Nash equilibrium fits only situations where the game is played often enough sothat a convention is settled. Our experiment should have been the ideal stage forGame Theory to "work". A simple and familiar game, very simple equilibriumstrategies, and repeated long enough to allow players to converge to equilibriumbehavior. Nevertheless, our findings hint at a systematic deviation from the GameTheoretical prediction. More importantly, it hints at the possibility that even in a simplesituation like matching pennies, the framing of the problem primes players to usedifferent procedures for playing the game, and that these procedures do not performequally well.

ReferencesAttali Yigal and Bar-Hillel Maya (2003). "Guess Where: The Position of Correct

Answers in Multiple-Choice Test Items as a Psychometric Variable". Journal ofEducational Measurement, 40, 109-128

Bargh, John A., Mark Chen and Lara Burrows (1996). "Automaticity of socialbehavior: Direct effects of trait construct and stereotype activation on action", Journalof Personality and Social Psychology 71, 230-244.

Bar-Hillel, Maya and Willem A. Wagenaar (1991). "The perception ofRandomness", Advances in Applied Mathematics 12, 428-454.

Ben-Porath, Elchanan (1993). "Repeated Games with Finite Automata" Journal ofEconomic Theory, 59, 17-32

Cooper, Russell, Douglas, V. DeJong, Robert Forstythe and Thomas, W. Ross,(1993). "Forward Induction in the Battle-of-the-Sexes Games", The American

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Economic Review, 83, 1303-1316.Guth Werner, Yaakov Kareev and Hartmut Kliemt (2005). "How to play randomly

without random generator - The case of maximin players", Homo Oeconomicus, 22,231-255

Kahneman, Daniel and Amos Tversky (1972). "Subjective Probability: A Judgmentof Representativeness", Cognitive Psychology 3, 430-454.

Mookherjee, Dilip and Barry Sopher (1994). "Learning Behavior in an ExperimentalMatching Pennies Game," Games and Economic Behavior 7, 62-91

Neyman, Abraham (1997). "Cooperation, Repetition and Automata" inCooperation: Game-Theoretic Approaches, NATO ASI Series F, Vol. 155, S. Hart andA. Mas-Colell (eds.), Springer-Verlag , pp. 233–255

Ochs, Jack (1995). "An Experimental Study of Games with Unique Mixed StrategyEquilibria", Games and Economic Behaviour 10, 202-217.

O’Neill, Barry (1987). "Non-Parametric Test of the Minimax Theory of Two-PersonZerosum Games" Proceedings of the National Academy of Sciences of the UnitedStates of America 84, 2106-2109

Rubinstein, Ariel (2001). "A Theorist’s View of Experiments", European EconomicReview 45, 615-628.

Rubinstein, Ariel, Amos Tversky and Dana Heller (1996). "Naive Strategies inZero-sum Games", in Understanding Strategic Interaction - Essays in Honor ofReinhard Selten, W.Guth et al. (editors), Springer-Verlag, 394-402.

Shachat, Jason and John Wooders (2001). “On The Irrelevance of Risk Attitudes inRepeated Two-Outcome Games”, Games and Economic Behavior 34, 342-363.

Simon, Richard, Paul E. Sly and Sivakumar Vilapakkam (1981). "Effect ofCompatibility of S-R Mapping on Reactions Towards the Stimulus Source", ActaPsychologica 47, 63-81

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Appendix

INSTRUCTIONS

IntroductionThis is an experiment in interactive decision-making. Various research institutes haveprovided the money for this experiment and you can make a considerable amount ofmoney in a short time if you pay attention.

The experimentYou are about to play a 24-round game. In the game, two players: a misleader and aguesser. At each round, the misleader moves first and has to choose a number, 0 or1. The guesser moves second and has to guess the misleader’s choice.

The guesser gets a point if his guess is correct and the misleader gets a point if theguesser fails. Each player’s aim is to get as many points as possible. At each roundeach player can see on the screen the entire history of the game up to that point.

The computer will randomly match the participants in this experiment into pairs. Ineach pair, one participant will be randomly assigned to play the role of the misleader,while the other participant will be assigned the role of the guesser.

PayoffsEach point that you earn in the game will be converted to $0.50. Your total payoff inthe experiment will be equal to the total sum of points you earn in the 24 periods of thegame, multiplied by $0.50, plus the show-up fee of $3. The minimum payoff is $3 andthe maximum is $15.

How to start the experiment?On the screen before you type in the following information:

Course Number: 1323

E-mail: Look at the number of the station, you are sitting in. If you are sitting in lab 1,the e-mail you should type in is 1lab1@nyu.edu. Similarly, if you are sitting in lab 2,your e-mail is 1lab2@nyu.edu. In general, if you are sitting in lab x, your e-mail is1labx@nyu.edu.

Password: If you are sitting in lab x, your password is 1x. For example, if you aresitting in lab 1, your password is 11. If you are sitting in lab 13, your password is 113,and so forth.

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